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  1. With the exception, perhaps, of some Lippmann emulsions quite recently investigated with the aid of the electron microscope; but even these show an approximately “normal” distribution of diameters, and thus ultimately of x rather than x.
  2. R. P. Loveland and A. P. H. Trivelli, “Mathematical methods of frequency analysis of size of particles,” J. Frank. Inst. 204, 193–214, 377–389 (1927).
    [Crossref]
  3. J. C. Kapteyn and M. J. Van Uven, Skew Frequency Curves in Biology and Statistics, second paper (Groningen, 1916).
  4. As a matter of fact, a law of error identical with (K) was derived a long time ago by Donald McAlister (Proc. Roy. Soc. 29, 365 (1879)) from the requirement that it should lead to the geometric mean as the most probable value of the measured quantity.
    [Crossref]
  5. At first a fair approach to constancy has been achieved in a few cases by taking |x−ξ| itself as a factor of φ, but the analysis of a larger experimental material has shown that ∣x-ξ∣12 as factor gives a much closer approximation to constancy.

1927 (1)

R. P. Loveland and A. P. H. Trivelli, “Mathematical methods of frequency analysis of size of particles,” J. Frank. Inst. 204, 193–214, 377–389 (1927).
[Crossref]

1879 (1)

As a matter of fact, a law of error identical with (K) was derived a long time ago by Donald McAlister (Proc. Roy. Soc. 29, 365 (1879)) from the requirement that it should lead to the geometric mean as the most probable value of the measured quantity.
[Crossref]

Kapteyn, J. C.

J. C. Kapteyn and M. J. Van Uven, Skew Frequency Curves in Biology and Statistics, second paper (Groningen, 1916).

Loveland, R. P.

R. P. Loveland and A. P. H. Trivelli, “Mathematical methods of frequency analysis of size of particles,” J. Frank. Inst. 204, 193–214, 377–389 (1927).
[Crossref]

McAlister, Donald

As a matter of fact, a law of error identical with (K) was derived a long time ago by Donald McAlister (Proc. Roy. Soc. 29, 365 (1879)) from the requirement that it should lead to the geometric mean as the most probable value of the measured quantity.
[Crossref]

Trivelli, A. P. H.

R. P. Loveland and A. P. H. Trivelli, “Mathematical methods of frequency analysis of size of particles,” J. Frank. Inst. 204, 193–214, 377–389 (1927).
[Crossref]

Van Uven, M. J.

J. C. Kapteyn and M. J. Van Uven, Skew Frequency Curves in Biology and Statistics, second paper (Groningen, 1916).

J. Frank. Inst. (1)

R. P. Loveland and A. P. H. Trivelli, “Mathematical methods of frequency analysis of size of particles,” J. Frank. Inst. 204, 193–214, 377–389 (1927).
[Crossref]

Proc. Roy. Soc. (1)

As a matter of fact, a law of error identical with (K) was derived a long time ago by Donald McAlister (Proc. Roy. Soc. 29, 365 (1879)) from the requirement that it should lead to the geometric mean as the most probable value of the measured quantity.
[Crossref]

Other (3)

At first a fair approach to constancy has been achieved in a few cases by taking |x−ξ| itself as a factor of φ, but the analysis of a larger experimental material has shown that ∣x-ξ∣12 as factor gives a much closer approximation to constancy.

J. C. Kapteyn and M. J. Van Uven, Skew Frequency Curves in Biology and Statistics, second paper (Groningen, 1916).

With the exception, perhaps, of some Lippmann emulsions quite recently investigated with the aid of the electron microscope; but even these show an approximately “normal” distribution of diameters, and thus ultimately of x rather than x.

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f = f ( x ) ,
f ( x i ) = x i - 1 2 Δ x i + 1 2 Δ g ( u ) d u .
1 Δ f ( x i ) - g ( x i ) = Δ 2 6 g ( x i ) ,
f ( x i ) - g ( x i ) = 1 6 g ( x i ) .
p.e. ( f ) = ± 0.477 [ 2 f ( n - f ) / n ] 1 2 .
± 0.477 / ( 2 n ) 1 2 .
f ( x ) = f m exp [ - c 2 ( x - x m ) 2 ] ,
f ( x ) = f m exp [ - c 2 ( log x - log x m ) 2 ] = f m exp [ - c 2 log 2 ( x / x m ) ] .
f ( x ) = f m exp ( - [ h ( x ) - h ( x m ) ] 2 ) ,
f ( x ) = f m · 10 - φ ( x ) ( x - x m ) 2 ,
φ ( x ) = - Log F ( x ) / ( x - x m ) 2 .
φ ( x ) · x - ξ 1 2 = κ constant .
φ ( x ) = κ / x - ξ 1 2 ,
f ( x ) = f m · 10 - κ ( x - x m ) 2 / x - ξ 1 2 .
f ( ξ ) = 0 ,
ξ - Δ / 2 ξ + Δ / 2 f ( x ) d x ,
x 1 = 1 3 ( 4 ξ - x m ) ;
x = 20 × mean class-size in μ 2 ,
f m = 300 at x m = 4 ,
ξ = 1.31 ,             κ = 0.0452 .
Log ( f / 300 ) = - 0.0452 ( x - 4 ) 2 / x - 1.31 1 2 .
n = 1035.
f m = 425 ,             at             x m = 2.
Log ( f / 425 ) = - 0.0621 ( x - 2 ) 2 / x - 3.94 1 2 .
n = 1010 .
Log ( f / 425 ) = - 0.070 ( x - 2 ) 2 / x - 1.21 1 2 .
n = 2819.
Log ( f / 450 ) = - 0.0923 ( x - 2 ) 2 / x - 1.77 1 2 .
n = Σ f = 717
x m = 3.50 ,             f m = 235.
ξ = 1.31 ,             κ = 0.0596 ,
Log ( f ( x ) / 235 ) = - 0.0596 ( x - 3.50 ) 2 / x - 1.31 1 2 .